Symmetry and Crystal System

Crystal may be distinguished from one another by their external morphology which generally appears to have some regularity of arangement. It is soon recognised that the regularity of external arragement implies a regularity of internal arrangement. This regularity is sometimes obscured by the presence of some over-grown or destroyed faces. Strictly speaking, classification of crystals is made on the basis of their characteristic internal structure which possesses some define symmetry. The symmetry of solid body may be defined in terms of the various elements of symmetry it possesses. Suppose we are looking at a cube and if for a moment we take our sight away from this cube and in the meantime someone rotates the cube through 90° about an imaginary axis passing through the centre of the top and bottom faces and if we look at the cube again, we will no find any change. This second orientation of crystal is exactly similar to the first orientation. In a similar way, the cube can be rotated twice more to get similar orientations. Such a crystal which has more than one orientation in space and which are indistinguishable from another is said to possess symmetry. The imaginary axis about which this rotation operation has been performed is a symmetry element, called axis of ratation. In this case the crystal has been rotated through 90° and so it possesses a 360/90 = 4-fold axis of rotation. Since a cube consists of six faces, there will be three 4-fold axis of rotation. (see the figure)

Similary, a cube may be brought into coincidence with itsef by rotation through one third of a revolution (120°) about a body diagonal. Such an axis of symmetry is called a 360/120 = 3-fold axis of symmetry. Since there are four body diagonals in a cube, it has four 3-fold axis of rotazion (see the figure). The figure depicts one of the six axes of 2-fold symmetry, emerging from opposite edges of a cube. The one centre of symmetry of a cube is represented simply at the mass centre (middle point) of a cube.

If we take a hexagon, we observe that it can be brought into coincidence with itself by rotation through 60°. So a hexagonal crystal posesses 360/60 = 6-fold axis of rotation.

a crystal may possess sevaral symmetry elements and these are:

1. axis of rotation,
2. axis of rotation-inversion,
3. plane of symmetry,
4. centre of inversion.

However, solids can be divided into sevendistinct types on the basis of only two of these symmetry elements. These are: axis of rotation, and axis of rotation-inverision.

An n-fold axis of rotation-inversion combines rotation through 360°/n with inversion through the center of symmetry of the crystal. Asoild of type shown in the figure possesses a four-fold axis of rotation-inversion. It is also represented as .

It can be shown that the onlyaxes of symmetry which are possible in crystals are 2-, 3-, 4- and 6-fold axis and it is found that the symmetry of all crystals can be described in terms of the following symmetry elements or combination thereof; 2-, 3-, 4- and 6- fold axes of simple rotation or axis of rotation-inversion,  and . For example, the 5-, 7-, 8- and higher fold symmetry does not occur in crystals. Why? The answer is that it is impossible to make a compact repeating pattern with these shapes. If we try to pack together some pentagons, we will soon find that some space is always left vacant. The only figures that can be packed together are the parallelogram (2-fold symmetry), the equilateral triangle (3-fold), the square (4-fold) and hexagon (6-fold). These are the only axial symmetries that are found in crystals. We may also add another axial symmetry, one-fold axis, which really means the absence of symmetry.